You can use a screwdriver but a brake adjusting tool works much better. They are very cheap, $3-$4 at any auto place.
To remove the drum, the brakes need to be loosened, usually not all the way, just enough so you can pull off the drum. Some trailer drums need a puller, like a gear puller. The one I know needs a puller is the Dexter EZ Lube. Most others just pull off with your hands though.